Domains of existence for finely holomorphic functions

Abstract

We show that fine domains in $\mathbf{C}$ with the property that they are Euclidean $F_\sigma$ and $G_\delta$, are in fact fine domains of existence for finely holomorphic functions. Moreover \emph{regular} fine domains are also fine domains of existence. Next we show that fine domains such as $\mathbf{C}\setminus \mathbf{Q}$ or $\mathbf{C}\setminus (\mathbf{Q}\times i\mathbf{Q})$, more specifically fine domains $V$ with the properties that their complement contains a non-empty polar set $E$ that is of the first Baire category in its Euclidean closure $K$ and that $(K\setminus E)\subset V$, are NOT fine domains of existence.